According to kepler's law of periods, `T^(2) = k r^(3)`, where `k` is a constant. Compute the constant `k` for (a) the earth and (b) the venus. Given that orbital radii of the earth and the venus are `1.496 xx 10^(11) m` and `1.082 xx 10^(11) m`, and their respective periods are 1 year and `0.615` year.

Explanation

Given `T^(2) = k r^(3)` or `k = T^(2)//r^(3)`
(a) For the earth, `r = 1.496 xx 10^(11) m` ,
`T = 1 year = 365 xx 24 xx 60 xx 60 s`
`:. K = ((365 xx 24 xx 60 xx 60)^(2))/((1.496 xx 106^(11))^(3))`
`= 2.97 xx 10^(-19) s^(2) m^(-3)`
(b) For the venus, `r = 1.082 xx 10^(11) m`,
`T = 0.615 year = 0.615 xx 365 xx 24 xx 60 xx 60 s`
`K = ((0.615xx 365xx 24xx 60 xx 60)^(2))/((1.082 xx 10^(11))^(3))`
`= 2.97 xx 10^(-19) s^(2) m^(-3)`